lab/exercicios_cap3
Felipe Albuquerque
18 de junho de 2019
Exercícios
1.
A hipótese nula indica que o orçamento para Televisão, Jornais e Rádio não impactam as vendas. Em notação: \(H_{0}^{1}: \beta_{1} = 0\), \(H_{0}^{2}: \beta_{2} = 0\), \(H_{0}^{3}: \beta_{3} = 0\). Para TV(\(\beta_{1}\)) e Rádio(\(\beta_{2}\)), podemos rejeitar a hipótese nula. Para jornais(\(\beta_{3}\)), não podemos.
2.
O KNN (K-nearest-neighbors) é um método de classificação de dados categóricos. Nele, se identifica os valores mais próximos de \(x_{0}\) e depois estima a probabilidade de \(x_{0}\) ser daquela categoria. O modelo de regressão KNN, da mesma forma, identifica os valores mais próximos de \(x_{0}\), e depois estima \(f(x_{0})\) como a média das respostas (na training data) entre esses “vizinhos”.
3.
A função fica \(Y = 50 + 20gpa + 0.07iq + 35gender + 0.01gpa*iq - 10gpa*gender\)
a
b
iq = 110
gpa = 4
gender = 1
Y = 50 + 20*gpa + 0.07*iq + 35*gender + 0.01*110*4 - 10*gpa*gender
Y## [1] 137.1c
Falso. O parâmetro da interação só mede a intensidade da interação. Para verificar se há um efeito significativo da interação, devemoz conduzir um teste de hipótese e olhar o p-valor.
4
a
A primeira vista, esperamos que a regressão linear tenha um RSS menor, já que a relação entre X e Y é linear.
b
c
Aqui acontece o oposto. A regressão polinomial é mais flexível que a linear. Como a relação entre X e Y é não-linear, esperamos portanto um menor RSS com a polinomial.
d
5
\[\hat{y_{i}} =x_{i}\hat{\beta}\] e:
\[\hat{\beta} = \frac{\sum_{i=1}^{n}x_{i}y_{i}}{\sum_{i^{'}=1}^{n}x_{i^{'}}^{2}}\] Então:
\[\hat{y_{i}} = x_{i}\frac{\sum_{i=1}^{n}x_{i}y_{i}}{\sum_{i^{'}=1}^{n}x_{i^{'}}^{2}}\]
logo:
\[\hat{y_{i}} = \sum_{i=1}^{n}(\frac{x_{i}y_{i} * x_{i}}{\sum_{i^{'}=1}^{n}x_{i^{'}}^{2}})\]
\[\hat{y_{i}} = \sum_{i=1}^{n}(\frac{x_{i} * x_{i}}{\sum_{i^{'}=1}^{n}x_{i^{'}}^{2}}y_{i})\] \[a_{i^{'}} = \frac{x_{i}*x_{i}}{\sum_{i^{'}=1}^{n}x_{i^{'}}^{2}}\]
6
7
8
a
auto = Auto
lm1 = lm(mpg ~ horsepower, data = auto)
summary(lm1)##
## Call:
## lm(formula = mpg ~ horsepower, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.5710 -3.2592 -0.3435 2.7630 16.9240
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.935861 0.717499 55.66 <0.0000000000000002 ***
## horsepower -0.157845 0.006446 -24.49 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.906 on 390 degrees of freedom
## Multiple R-squared: 0.6059, Adjusted R-squared: 0.6049
## F-statistic: 599.7 on 1 and 390 DF, p-value: < 0.00000000000000022i Há uma associação clara entre horsepower e mpg.
ii o p valor é menor que \(2e^{-16}\), o que indica uma associação forte.
iii associação negativa, já que o coeficiente é de \(-0.157\).
iiii Para horsepower - 98, valor de mpg =
hpr98 <- data.frame(horsepower=98)
predict(lm1, hpr98)## 1
## 24.46708Intervalo de confiança:
predict(lm1, hpr98, interval = "confidence")## fit lwr upr
## 1 24.46708 23.97308 24.96108b
auto %>%
ggplot(aes(x = horsepower, y = mpg)) + geom_point() +
geom_smooth(method = "lm") +
theme_hc()
c
par(mfrow=c(2,2))
plot(lm1)
Há uma não-linearidade dos resíduos.
9
a
auto %>%
dplyr::select(-name) %>%
GGally::ggpairs()
b
auto %>%
dplyr::select(1:8) %>%
cor()## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7776175 -0.8051269 -0.7784268 -0.8322442
## cylinders -0.7776175 1.0000000 0.9508233 0.8429834 0.8975273
## displacement -0.8051269 0.9508233 1.0000000 0.8972570 0.9329944
## horsepower -0.7784268 0.8429834 0.8972570 1.0000000 0.8645377
## weight -0.8322442 0.8975273 0.9329944 0.8645377 1.0000000
## acceleration 0.4233285 -0.5046834 -0.5438005 -0.6891955 -0.4168392
## year 0.5805410 -0.3456474 -0.3698552 -0.4163615 -0.3091199
## origin 0.5652088 -0.5689316 -0.6145351 -0.4551715 -0.5850054
## acceleration year origin
## mpg 0.4233285 0.5805410 0.5652088
## cylinders -0.5046834 -0.3456474 -0.5689316
## displacement -0.5438005 -0.3698552 -0.6145351
## horsepower -0.6891955 -0.4163615 -0.4551715
## weight -0.4168392 -0.3091199 -0.5850054
## acceleration 1.0000000 0.2903161 0.2127458
## year 0.2903161 1.0000000 0.1815277
## origin 0.2127458 0.1815277 1.0000000c
lm2 <- lm(mpg ~ . -name, data = auto)
summary(lm2)##
## Call:
## lm(formula = mpg ~ . - name, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5903 -2.1565 -0.1169 1.8690 13.0604
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.218435 4.644294 -3.707 0.00024 ***
## cylinders -0.493376 0.323282 -1.526 0.12780
## displacement 0.019896 0.007515 2.647 0.00844 **
## horsepower -0.016951 0.013787 -1.230 0.21963
## weight -0.006474 0.000652 -9.929 < 0.0000000000000002 ***
## acceleration 0.080576 0.098845 0.815 0.41548
## year 0.750773 0.050973 14.729 < 0.0000000000000002 ***
## origin 1.426141 0.278136 5.127 0.000000467 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.328 on 384 degrees of freedom
## Multiple R-squared: 0.8215, Adjusted R-squared: 0.8182
## F-statistic: 252.4 on 7 and 384 DF, p-value: < 0.00000000000000022i Existe
ii displacement (0.019896) ; weight (-0.006474); year (0.750773) e origin (1.426141)
iii versões mais recentes de carros conseguem consumir menos combustível
d
par(mfrow=c(2,2))
plot(lm2)
Os erros estão mais lineares, o que aponta para uma homocedasticidade desejada.
e
plot(auto$year, auto$horsepower)
lm3 <- lm(mpg ~ horsepower + year + horsepower*year, data = auto)
summary(lm3)##
## Call:
## lm(formula = mpg ~ horsepower + year + horsepower * year, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.3492 -2.4509 -0.4557 2.4056 14.4437
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -126.608853 12.117256 -10.449 <0.0000000000000002 ***
## horsepower 1.045674 0.115374 9.063 <0.0000000000000002 ***
## year 2.191976 0.161350 13.585 <0.0000000000000002 ***
## horsepower:year -0.015959 0.001562 -10.217 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.901 on 388 degrees of freedom
## Multiple R-squared: 0.7522, Adjusted R-squared: 0.7503
## F-statistic: 392.5 on 3 and 388 DF, p-value: < 0.00000000000000022par(mfrow=c(2,2))
plot(lm3)
f
lm4 <- lm(mpg ~ weight, data = auto)
summary(lm4)##
## Call:
## lm(formula = mpg ~ weight, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.9736 -2.7556 -0.3358 2.1379 16.5194
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.216524 0.798673 57.87 <0.0000000000000002 ***
## weight -0.007647 0.000258 -29.64 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.333 on 390 degrees of freedom
## Multiple R-squared: 0.6926, Adjusted R-squared: 0.6918
## F-statistic: 878.8 on 1 and 390 DF, p-value: < 0.00000000000000022par(mfrow = c(2,2))
plot(lm4)
lm5 <- lm(mpg ~ log(weight), data = auto)
summary(lm5)##
## Call:
## lm(formula = mpg ~ log(weight), data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.4315 -2.6752 -0.2888 1.9429 16.0136
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 209.9433 6.0002 34.99 <0.0000000000000002 ***
## log(weight) -23.4317 0.7534 -31.10 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.189 on 390 degrees of freedom
## Multiple R-squared: 0.7127, Adjusted R-squared: 0.7119
## F-statistic: 967.3 on 1 and 390 DF, p-value: < 0.00000000000000022par(mfrow = c(2,2))
plot(lm5)
Normalizando a variável weight, os erros ficam praticamente lineares. _____________________________ # 10
rm(list=ls())
data = Carseatsa
lm1 <- lm(Sales ~ Population + Urban + US, data = data)
summary(lm1)##
## Call:
## lm(formula = Sales ~ Population + Urban + US, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.3323 -1.9844 -0.0824 1.8783 8.4053
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.7262086 0.4009409 16.776 < 0.0000000000000002 ***
## Population 0.0007415 0.0009499 0.781 0.435475
## UrbanYes -0.1341034 0.3063701 -0.438 0.661830
## USYes 1.0360741 0.2921241 3.547 0.000437 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.787 on 396 degrees of freedom
## Multiple R-squared: 0.03342, Adjusted R-squared: 0.02609
## F-statistic: 4.563 on 3 and 396 DF, p-value: 0.003713b
O aumento de uma unidade da população aumenta em 0.07 as unidades vendidas (0.0007);
Em áreas urbanas, as vendas são menores em 22 mil unidades (-0.0219)
As vendas aumentam em até 1000 unidades em locais dentro dos Estados Unidos (1.0360)
c
\[Sales = 6.7626 \ + 0.0007(Population)\ - 0.1341(Urban)\ + 1.0360(US)\] ## d
Pode-se rejeitar a hipótese nula para USYes.
e
lm2 <- lm(Sales ~ US, data = data)
summary(lm2)##
## Call:
## lm(formula = Sales ~ US, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.497 -1.929 -0.105 1.836 8.403
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.8230 0.2335 29.21 < 0.0000000000000002 ***
## USYes 1.0439 0.2908 3.59 0.000372 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.783 on 398 degrees of freedom
## Multiple R-squared: 0.03136, Adjusted R-squared: 0.02893
## F-statistic: 12.89 on 1 and 398 DF, p-value: 0.0003723f
O segundo modelo e o primeiro contam um \(X^{2}\) muito pequeno, de apenas 0.02. O RSE do segundo modelo é um pouco menor.
g
confint(lm2)## 2.5 % 97.5 %
## (Intercept) 6.3638993 7.282157
## USYes 0.4721887 1.615553h
outlierTest(lm1)## No Studentized residuals with Bonferroni p < 0.05
## Largest |rstudent|:
## rstudent unadjusted p-value Bonferroni p
## 377 3.05528 0.0024011 0.9604311
rm(list=ls())
set.seed(1)
x <- rnorm(100)
y <- 2*x + rnorm(100)a
lm1 <- lm(y ~ x -1)
summary(lm1)##
## Call:
## lm(formula = y ~ x - 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9154 -0.6472 -0.1771 0.5056 2.3109
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## x 1.9939 0.1065 18.73 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9586 on 99 degrees of freedom
## Multiple R-squared: 0.7798, Adjusted R-squared: 0.7776
## F-statistic: 350.7 on 1 and 99 DF, p-value: < 0.00000000000000022b
lm2 <- lm(x ~ y -1)
summary(lm2)##
## Call:
## lm(formula = x ~ y - 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.8699 -0.2368 0.1030 0.2858 0.8938
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## y 0.39111 0.02089 18.73 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4246 on 99 degrees of freedom
## Multiple R-squared: 0.7798, Adjusted R-squared: 0.7776
## F-statistic: 350.7 on 1 and 99 DF, p-value: < 0.00000000000000022c
Pelo visto, obtemos os mesmo valores para a estatística-t, e consequentemente, para o p-valor. Em outras palavras, \(y = 1.99x + \epsilon\) é igual a \(x = 0.39y + \epsilon\).
d
e
f
lm3 <- lm(y ~ x)
summary(lm3)##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8768 -0.6138 -0.1395 0.5394 2.3462
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.03769 0.09699 -0.389 0.698
## x 1.99894 0.10773 18.556 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9628 on 98 degrees of freedom
## Multiple R-squared: 0.7784, Adjusted R-squared: 0.7762
## F-statistic: 344.3 on 1 and 98 DF, p-value: < 0.00000000000000022##
## Call:
## lm(formula = x ~ y)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.90848 -0.28101 0.06274 0.24570 0.85736
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.03880 0.04266 0.91 0.365
## y 0.38942 0.02099 18.56 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4249 on 98 degrees of freedom
## Multiple R-squared: 0.7784, Adjusted R-squared: 0.7762
## F-statistic: 344.3 on 1 and 98 DF, p-value: < 0.00000000000000022De novo, ambas estatísticas-t se assemelham.
12
a
Se \(\hat{\beta} = \frac{\sum_{i}x_{i}y_{i}}{\sum_{j}x_{j}^{2}}\) e \(\hat{\beta^{'}} = \frac{\sum_{i}x_{i}y_{i}}{\sum_{j}y_{j}^{2}}\), então os coeficientes são iguais se:
\[\sum_{j}x_{j}^{2} = \sum_{j}y_{j}^{2}\]
b
c
13
a
rm(list=ls())
set.seed(1)
x <- rnorm(1000, 0, 1)b
eps <- rnorm(1000, 0, 0.25)c
y <- -1 + 0.5*x + eps # eps=epsilon=e
length(y)## [1] 1000d
data = as.data.frame(cbind(x, y))
data %>%
ggplot(aes(x = x, y = y)) + geom_point() + theme_hc()
Parece haver uma correlação positiva e linear entre as variáveis
e
lm1 <- lm(y ~ x)
summary(lm1)##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.81211 -0.16799 -0.00344 0.18885 0.91108
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.004047 0.008226 -122.05 <0.0000000000000002 ***
## x 0.501608 0.007952 63.08 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2601 on 998 degrees of freedom
## Multiple R-squared: 0.7995, Adjusted R-squared: 0.7993
## F-statistic: 3979 on 1 and 998 DF, p-value: < 0.00000000000000022Os parâmetros se assemelham.
f
data %>%
ggplot(aes(x = x, y = y)) + geom_point() +
geom_smooth(method = "lm") + geom_abline(aes(intercept = -1, slope = 0.5), col = "red") +
theme_hc()
g
lm2 <- lm(y ~ x + I(x^2))
summary(lm2)##
## Call:
## lm(formula = y ~ x + I(x^2))
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.82236 -0.16995 -0.00384 0.18910 0.90073
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.009147 0.010077 -100.147 <0.0000000000000002 ***
## x 0.501814 0.007957 63.069 <0.0000000000000002 ***
## I(x^2) 0.004768 0.005440 0.877 0.381
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2602 on 997 degrees of freedom
## Multiple R-squared: 0.7996, Adjusted R-squared: 0.7992
## F-statistic: 1989 on 2 and 997 DF, p-value: < 0.00000000000000022Não. Pelo contrário.
h
eps2 <- rnorm(1000, 0, 0.1)
y2 <- -1 + 0.5*x +eps2
lm3 <- lm(y2 ~ x)
summary(lm3)##
## Call:
## lm(formula = y2 ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35818 -0.06414 -0.00179 0.06866 0.28105
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.998412 0.003259 -306.4 <0.0000000000000002 ***
## x 0.504919 0.003150 160.3 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.103 on 998 degrees of freedom
## Multiple R-squared: 0.9626, Adjusted R-squared: 0.9626
## F-statistic: 2.569e+04 on 1 and 998 DF, p-value: < 0.00000000000000022data %>%
ggplot(aes(x = x, y = y2)) + geom_point() +
geom_smooth(method = "lm") + geom_abline(aes(intercept = -1, slope = 0.5), col = "red") +
theme_hc()
i
eps3 <- rnorm(1000, sd=3) # orig sd was 0.5
y3 <- -1 + 0.5*x + eps3
lm4 <- lm(y3 ~ x)
summary(lm4)##
## Call:
## lm(formula = y3 ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.6046 -2.0975 -0.0328 2.1509 9.1657
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.94901 0.09856 -9.628 < 0.0000000000000002 ***
## x 0.57067 0.09528 5.989 0.00000000294 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.117 on 998 degrees of freedom
## Multiple R-squared: 0.0347, Adjusted R-squared: 0.03373
## F-statistic: 35.87 on 1 and 998 DF, p-value: 0.000000002937data %>%
ggplot(aes(x = x, y = y3)) + geom_point() +
geom_smooth(method = "lm") + geom_abline(aes(intercept = -1, slope = 0.5), col = "red") +
theme_hc()
O intervalo de confiança aumenta. O R² diminui consideravelmente, devido a grande variabilidade dos dados.
j
confint(lm1)## 2.5 % 97.5 %
## (Intercept) -1.0201895 -0.9879040
## x 0.4860032 0.5172131confint(lm3)## 2.5 % 97.5 %
## (Intercept) -1.0048061 -0.9920175
## x 0.4987378 0.5111003confint(lm4)## 2.5 % 97.5 %
## (Intercept) -1.1424261 -0.7555941
## x 0.3836967 0.7576412Quanto menor a variância, menor o intervalo de confiança.
_______________________________________________- # 14
a
rm(list=ls())
set.seed(1)
x1 = runif(100)
x2 = 0.5*x1 + rnorm(100)/10
x3 = 2 + 2*x1 + 0.3*x2 + rnorm(100)\(\beta_{0} = 2\), \(\beta_{1} = 2\), \(\beta_{2} = 0.3\)
b
data = as.data.frame(cbind(x1, x2))
cor(x1, x2)## [1] 0.8351212data %>%
ggplot(aes(x = x1, y = x2)) + geom_point() + theme_hc()
c
lm1 <- lm(x3 ~ x1 + x2)
summary(lm1)##
## Call:
## lm(formula = x3 ~ x1 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8311 -0.7273 -0.0537 0.6338 2.3359
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.1305 0.2319 9.188 0.00000000000000761 ***
## x1 1.4396 0.7212 1.996 0.0487 *
## x2 1.0097 1.1337 0.891 0.3754
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.056 on 97 degrees of freedom
## Multiple R-squared: 0.2088, Adjusted R-squared: 0.1925
## F-statistic: 12.8 on 2 and 97 DF, p-value: 0.00001164Coeficiente \(\beta_{1}\) = 1.43 e Coefiencie \(\beta_{2}\) = 1.0097. A hipótese nula pode ser rejeitada a 5% no primeiro caso, mas não no segundo.
d
lm2 <- lm(x3 ~ x1)e
Nos dois casos daria para rejeitar a hipótese nula
f
Não. Isso acontece apenas pela multicolinearidade entre as variáveis x1 e x2.
g
x1=c(x1,0.1)
x2=c(x2,0.8)
y=c(x3,6)lm4 <- lm(y ~ x1 + x2)
lm5 <- lm(y ~ x1)
lm6 <- lm(y ~ x2)par(mfrow=c(2,2))
plot(lm4)
par(mfrow=c(2,2))
plot(lm5)
par(mfrow=c(2,2))
plot(lm6)
15
rm(list=ls())
boston = Boston
attach(boston)a
Isoladamente, todas as variáveis são significativas, menos chas
b
lm1 <- lm(crim ~ ., data = boston)
summary(lm1)##
## Call:
## lm(formula = crim ~ ., data = boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.924 -2.120 -0.353 1.019 75.051
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.033228 7.234903 2.354 0.018949 *
## zn 0.044855 0.018734 2.394 0.017025 *
## indus -0.063855 0.083407 -0.766 0.444294
## chas -0.749134 1.180147 -0.635 0.525867
## nox -10.313535 5.275536 -1.955 0.051152 .
## rm 0.430131 0.612830 0.702 0.483089
## age 0.001452 0.017925 0.081 0.935488
## dis -0.987176 0.281817 -3.503 0.000502 ***
## rad 0.588209 0.088049 6.680 0.0000000000646 ***
## tax -0.003780 0.005156 -0.733 0.463793
## ptratio -0.271081 0.186450 -1.454 0.146611
## black -0.007538 0.003673 -2.052 0.040702 *
## lstat 0.126211 0.075725 1.667 0.096208 .
## medv -0.198887 0.060516 -3.287 0.001087 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.439 on 492 degrees of freedom
## Multiple R-squared: 0.454, Adjusted R-squared: 0.4396
## F-statistic: 31.47 on 13 and 492 DF, p-value: < 0.00000000000000022Podemos rejeitar a hipótese nula a 5% para **zn, dis, rad, black, medv